It is okay to wear summer clothes. I suggest you take a light jackt in case of cool temperature in Lijiang for its high altitude and in hotels for the air conditioner. Showers will be not a big problem for your stay in Guilin. In fact, summer in those area are very cool. Usually, the shower will last a coupld of hours only. The scenery look amazing after rain. Guilin and Yangshuo are closed to each other. As to Lijiang, I think you should take airplane from Guilin. Or you can get to Kunming first and then take bus to Lijiang. If you want a tour, how many days do you plan to spend over these regions? You can read about this route itinerary that I think is a good reference for you.
https://www.travelchinaguide.com/tour/memorable/tour-111b.htm

Thank you for the info-s Mr Roseknight!